Nima Arkani-Hamed et al- Rapid asymmetric inflation and early cosmology in theories with...

8/3/2019 Nima Arkani-Hamed et al- Rapid asymmetric inflation and early cosmology in theories with sub-millimeter dimensio…

http://slidepdf.com/reader/full/nima-arkani-hamed-et-al-rapid-asymmetric-inflation-and-early-cosmology-in 1/49

SLAC-PUB-8068

Rapid asymmetric inflation and early cosmology intheories with sub-millimeter dimensions

Work supported by Department of Energy contract DE AC03 76SF00515.

Nima Arkani-Hamed et al.

March 1999

Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309

Physical Review DSubmitted to



1 Introduction

It was recently pointed out that the fundamental Planck mass could be close to theTeV scale[1, 2, 3, 4], thus providing a novel solution to the hierarchy problem for thestandard model. Gravity becomes comparable in strength to the other interactions atthis scale, and the observed weakness of gravity at long distances is then explained bythe presence of n new “large” spatial dimensions. Gauss’ Law relates the Planck scalesof the (4+ n) dimensional theory, M

∗, and the long-distance 4-dimensional theory, M pl ,

M 2pl = ( b0)n M n +2∗

(1)

where b0 is the (present, stabilized) size of the extra dimensions. If we put M ∗∼1 TeV

thenb0∼10−17+ 30

n cm (2)

For n = 1, b0∼10

13cm, so this case is excluded since it would modify Newtonian

gravitation at solar-system distances. Already for n = 2, however, b0∼1 mm, whichhappens to be the distance where our present experimental knowledge of gravitationalstrength forces ends. For larger n, 1/b 0 slowly approaches the fundamental Planckscale M

∗.

While the gravitational force has not been measured beneath a millimeter, the successof the SM up to ∼100GeV implies that the SM elds can not feel these extra largedimensions; that is, they must be stuck on a 3-dimensional wall, or “3-brane”, in thehigher dimensional space. Thus, in this framework the universe is (4 + n)-dimensional

with fundamental Planck scale near the weak scale, with n ≥ 2 new sub-mm sizeddimensions where gravity, and perhaps other elds, can freely propagate, but wherethe SM particles are localized on a 3-brane in the higher-dimensional space. The mostattractive possibility for localizing the SM elds to the brane is to employ the D-branesthat naturally occur in type I or type II string theory [ 5, 2]. Gauge and other degreesof freedom are naturally conned to such D-branes [ 5], and furthermore this approachhas the obvious advantage of being formulated within a consistent theory of gravity.However, from a practical point of view, the most important question is whether thisframework is experimentally excluded. This was the subject of [ 3] where laboratory,

astrophysical, and cosmological constraints were studied and found not to exclude theseideas.

There are a number of other papers discussing related suggestions. Refs. [ 6] examinethe idea of lowering the gauge-coupling unication scale by utilizing higher dimen-sions. Further papers concern themselves with the construction of string models with

1



extra dimensions larger than the string scale [ 7, 8, 9], gauge coupling unication inhigher dimensions without lowering the unication scale [ 10], the effective theory of the low energy degrees of freedom in realizations of our world as a brane [ 11] and radiusstabilization [ 12, 4]. There have also been many recent papers discussing various theo-

retical and phenomenological aspects of this scheme [ 13], and a few papers on aspectsof the early universe cosmology [14][15][16][17][18] that discuss issues related to thoseconsidered here.∗

In this paper we will discuss ination and other general aspects of early universecosmology in world-as-a-brane models. In particular we will be concerned with aspectsof early universe cosmology that involve the dynamics of the internal dimensions in acentral way. We nd that there exist attractive models of ination that occur whilethe internal dimensions are still small, far away from their nal stabilized value givenby the Gauss’ law constraint. We show that in such models it is very easy to produce

the required number of efoldings of ination even though no energy densities exceedthe fundamental Planck scale, ρ < M 4

∗, and that supersymmetry, if it exists at all, is

very badly broken on our brane. Further, we demonstrate that the density perturba-tions δρ/ρ as measured by COBE and the other microwave background and large-scalestructure experiments can be easily reproduced in such models, both in their magnitudeand in their approximately scale-invariant spectrum. In the most minimal approach,the inaton eld may be just the moduli describing the size of the internal dimensionsitself (the radion eld of [4]), the role of the inationary potential being played bythe stabilizing potential of the internal space. In the case of a wall-localized inaton,

the cosmological constant might even result from the electroweak phase transition, inwhich case the inaton is the Higgs. Actually, an important remark in this regard isthat when the internal dimensions are small, b∼M −1

∗, the distinction between on-the-

wall and off-the-wall physics is not meaningful: e.g. , the inationary features in V (b)at small b could be due to Higgs physics on the wall.

The approximately scale-invariant nature of the primordial perturbation spectrum∗ However, we note that a closer scrutiny of [ 18] reveals that it is impossible to get any ination

at all in the models considered. The potentials in [ 18] all have exponential dependence on the radioneld, and thus lead to the power-law dependence of the scale factor on time, a ∼t2/β 2

[21]. Thecorrect formula for the parameter β of [18] is β = 2

2n/ (n + 2), and thus β

≥2 for all cases n

≥2,

which gives only subluminal expansion, and not ination. In fact, a complete analysis must employthe physically meaningful measure of scales, namely the scale factor expressed in units of the Comptonwavelength of wall particles. (Because of the possibility of Weyl rescaling between string and Einsteinframes one must be careful about the denition of physical quantities.) In such physical units it ispossible to show that the scale factor always expands sub-luminally, even more slowly than the naiveEinstein frame expansion, and thus none of the solutions of [ 18] contain a stage of ination.

2



implies that, during ination, the internal dimensions must expand more slowly thanthe universe on the wall. Thus we are led to consider a form of asymmetric inationaryexpansion of the higher-dimensional world. It is very interesting to note that requiringthe consistency and naturalness of both the long duration of ination ( i.e. , number of

efolds N e >∼100), and the magnitude of δρ/ρ , implies that ination should occur early,with the internal dimensions close to their natural initial size bi∼M −1∗

. In particular,we avoid the introduction of extremely light inatons which seem to be needed if ination occurs after the internal dimensions reach their current, large, size [ 14, 15].We emphasize here that this constraint emerged from the assumption that duringination the internal space was already large and stable. In fact, in such scenariosobtaining δρ/ρ requires that the inaton is even lighter [ 16], and further, that inationoccurring only after stabilization cannot explain the age of the universe [ 16]. Indeed,the wall-only ination cannot begin before t ∼H −1

∼M pl /M 2∗

>> M −1∗

, when the

universe is already very large and old.However, in the models of ination at very early times which we consider, the effective

4D Planck mass is both much smaller than it is now, and in general, time-dependentdue to the variation in the volume of the internal dimensions. Moreover, inationoccurs soon after the “birth” of the universe when the sizes of all dimensions are closeto their natural initial size ∼M −1

∗. Hence early asymmetric ination solves the age

problem too.The framework in which we discuss these issues is that of semiclassical (4 + n)-

dimensional gravity with an additional potential V (b) depending on the size of the

internal dimensions. As discussed in Ref.[ 4] this rapidly becomes a good approximationat energy scales below M

∗, which, self consistently, is the correct regime for early

ination, essentially because of the COBE constraint that the density perturbations aresmall δρ/ρ ∼2×10−5 . Most of our analysis will utilize the so-called string frame withexplicit scale factors ( a(t), b(t)) for the sizes of our brane-localized dimensions and theinternal dimensions respectively. In this frame the effective long-distance 4-dimensionalvalue of Newton’s constant depends on b(t) and changes with time. However this isthe frame where the “measuring sticks” of particle masses and Compton wavelengthsare xed. On the other hand, in the analysis that follows we will nd it occasionally

useful to employ the correspondence of the low-energy (4+ n)-dimensional theory withan effective 4-dimensional theory which takes the form of a scalar-tensor theory of gravity, the radius of the large internal space b playing a role similar to the Brans-Dicke scalar eld. The inationary dynamics from the point of view of this frameis completely equivalent to the usual slow-roll scenarios, and the conditions for the

3



asymmetric expansion of the universe on the brane relative to the internal dimensionsare completely equivalent to the slow roll conditions in the usual inationary models.Further, this analogy provides very useful consistency check for the determinationof the density perturbations. The relationship between the scale factor b(t) and the

canonically normalized 4-dimensional (“radion”) eld is given in equation ( 23). (Wewill often abuse our terminology and refer to b(t) itself as the “radion”.)It is important to realize that under quite general conditions, the early inationary

era is followed by a long epoch where the scale factor of our brane-universe undergoesa slow contraction while the internal dimensions continue to expand towards their nalstabilized value. We show that even with the inclusion of a potential for b, it is possibleto exactly describe the evolution during this epoch, and we present a class of exactsolutions which generalize the usual vacuum Kasner solutions. The total amount of contraction of our universe on the brane is bounded by a small power of the expansion

of the size of internal dimensions, and varies between at most 7 efoldings in the case of two extra dimensions to at most 12 efoldings when there are six extra dimensions. Weshow that during this phase of b(t) evolution to the stabilization point, the productionof bulk gravitons by the time-varying metric remains completely suppressed, ensuringthat the bulk is very cold at, and after, the stabilization of the internal dimensions.

However, the particles produced by gravitational (Hawking) effects on the brane atthe end of the inationary de Sitter phase can play an important role. Their energydensity is blue-shifted by the slow contraction of a(t), and when this wall-localizedenergy density exceeds the energy density in the radion, the contraction of a(t) ceases,

with a “Big Bounce” occurring, leading on to an expansion of both a and b. It is verytempting to view the reheating on the wall as due to this contraction of primordialde Sitter radiation. This is especially so as the nal temperatures come naturally closeto the normalcy bounds (the upper bound T

∗on the temperature of [ 3] that ensure

that the evaporation to bulk gravitons remains negligible).However, such a conclusion is premature, since as we argue below the energy density

in the radion is naturally of the same order as that on the wall, and this leads to aradion moduli problem in the later evolution of the universe. Explicitly, because of the gravitationally suppressed couplings of this eld to wall-localized SM states, the

radion decays very slowly back into light wall elds. As a result, its energy densitywould red-shift only according to ρb∼1/a 3 compared to ρwall ∼1/a 4, and comes todominate the total energy density. This is just the standard moduli problem, which isinextricably linked with the mechanism of reheating in world-as-a-brane models.

4



Concretely, the basic picture we advocate is:

• The quantum creation of the universe takes place with the initial size of alldimensions close to the fundamental Planck scale M −1

∗. In particular the initial

size bi of the internal dimensions is of this size.

• A prolonged period of ination in a direction parallel to our brane takes place,with our scale factor a(t) increasing superluminally a(t)∼t p with p >> 1, andb(t) = bI essentially static. (In the cases that we explicitly discuss the ination isquasi-exponential, a(t)∼a i exp(Ht ).) This brane ination is driven by either thestabilizing potential of the radion b itself, or by a wall-localized eld with effectivenon-zero cosmological constant. In either case it is unnatural for the size of theeffective 4-dimensional cosmological constant to exceed roughly ( TeV) 4. Sincethe internal dimensions are small the effective 4-dimensional Newton’s constant

is largeGN,initial =

1bn

I M n +2∗

1M 2∗

. (3)

Thus the Hubble constant during this initial period of ination can be large eventhough the energy density is quite small, V ∼O( TeV 4),

H 2inV

bnI M n +2∗

V M 2∗

. (4)

Thus ination can be rapid , and moreover, as we will argue in detail the densityperturbations can be large, being determined in order of magnitude to be

δρρ

H in

M ∗(M ∗bI )n/ 2S

. (5)

where S is a parameter that encapsulates both the duration of a(t) ination andthe deviation of the perturbation spectrum from the scale-invariant Harrison-Zeldovich spectrum. (We will argue that S <

∼1/ 50.) We can turn this around by

imposing the COBE-derived normalization on δρ/ρ , and thus on V /b nI M n +2∗

.During this period the size bI of the internal radii are xed by the over-dampingarising from H in .

• Wall ination now ends, with H starting to drop and simultaneously the radionstarting to evolve to its’ minimum at b0. The initial motion of the radion can bequite involved, but from the coupled equations of motion for the scale factors a(t)and b(t) we will see that as b evolves towards its’ minimum, our scale factor, under

5



very general conditions, undergoes a collapse. The detailed behavior generalizesthe well-known Kasner solutions where some dimensions expand, while otherscollapse, both with determined (subluminal) power-law dependence on time.

•The contraction epoch ends when the blue-shifting radiation density on our branebecomes equal to the energy density in the radion eld ρwall = ρb. A model-independent initial source for the radiation on the wall is the Hawking radiationleft over from the early inationary de Sitter phase. After the contraction of a(t)reverses, the radion and wall-localized energy densities scale together until thestabilization point is reached.

• Finally around the stabilization point b0 the radion eld starts to oscillate freely.Since this energy density scales as 1 /a 3, and the wall-to-radion energy densitiesare initially comparable at the start of the oscillation era, the radion energy starts

to dominate the total energy density.

• The most serious question that early universe cosmology presents in the world-as-a-brane scenario is how do we dilute this energy in radion oscillations to anacceptable level. The radion is long-lived, its’ decay width back to light wallstates being given by †

Γϕm3ϕ

M 2pl. (6)

We thus require some dilution in the radion energy density, either by a shortperiod of late ination followed by reheating, or by a delayed reheating after ρb

has sufficiently red-shifted. The amount of dilution of the radion energy densitythat we require is given roughly by 1 eV /T

∗∼10−7 , so that only about 5 efoldsof late ination is needed.

The paper is organized as follows. In Section 2 we will discuss the details of theinationary stage, both in the original string (brane) frame and in the effective Einsteinframe, showing the initial conditions, the conditions on inationary potentials and theresolution of the COBE constraints. Section 3 is devoted to the study of the era of radion evolution to stabilization. In subsection 3.1 we derive the exact solutions which

generalize simple vacuum Kasner solutions, and which demonstrate that a period of †Here mϕ is the mass of the canonically normalized eld corresponding to b. The experimental

bound on this mass is mϕ >∼

10− 3 eV. Note that the decay width is increased if we take into accountthe possible presence of many branes in the bulk. Indeed some scenarios of radius stabilization e.g. ,the “brane crystallization” picture [ 4], require N wall (M pl /M ∗)2( n − 2) /n branes in the bulk. If eachof these have O(1) light modes then the total decay width to all branes is greatly enhanced.

6



a(t) contraction subsequent to ination is quite generic. In subsection 3.2 we discussthe phenomenological consequences of this era, including the cause of its cessation andthe (minimal) production of bulk gravitons. There we will also show how the normalcybounds for the temperature of our brane after stabilization are met by the evolution.

In subsection 3.3 we discuss we discuss the issue of the production of bulk gravitonsduring the era of radion evolution and show that the quantities produced are harmless.In subsection 3.4 we provide a physically quite useful discussion of the contraction andstabilization eras in the Einstein frame, where many of the arguments of the previoussubsections can be understood quite simply. In subsection 3.5 we provide a shortdiscussion of the radion moduli problem, and quantify its’ size using the evolutionequations discussed earlier in Section 3. Section 4 gives our conclusions. We alsoinclude three extensive Appendices: Appendix A contains a short discussion of some of the basic kinematics of brane evolution embedded in higher dimensions. Appendix B

gives some additional details of the exact solutions for the post-ination slow evolutionera where a(t) contracts. Appendix C proves that the contraction of a(t) is self-ceasingby the blue-shifting of brane-localized radiation, and also discusses the exact solutionsthat may be obtained for the “Big Bounce” that terminates this contraction.

2 Early ination

We now embark on our detailed discussion of the evolution of the sizes of our braneand the transverse “internal” dimensions. In particular in this section we focus on the

physics of the (early) inationary epoch.We are interested in the case of a 3-brane embedded in a (4 + n)-dimensional space-

time. To reproduce our observed world, ultimately the 3 spatial dimensions parallel tothe brane must be as least as large as our current horizon size, while the n transversespatial dimensions have to stabilize at a size b0 given by the constraint on GN , (1),(b0)n M n +2

∗= M 2pl . Note that we take the internal space to be topologically compact

from the beginning. That is we impose periodic boundary conditions in the directionstransverse to the wall. These conditions reect the fact that the low energy theory isfour-dimensional, and that the classical evolution preserves the topology.

The total action is comprised of a bulk part,

S bulk = − d4+ n x −det G(4+ n ) M (n +2)∗ R −Lmatter + . . . , (7)

7



and a brane part,

S brane = − d4x −det ginduced(4) Lstandard model + . . . , (8)

where

Lmatter is the Lagrangian of the bulk elds apart from the graviton. These elds

give rise to the stabilizing potential V that we discuss below. The ellipses denotehigher-derivative terms that can be safely ignored in most of the regime of interestas curvatures are small compared to the fundamental Planck scale M

∗, apart possibly

from the very early pre-inationary stage immediately after the quantum creation of the universe. However any signatures of such a phase of high energy and large curvaturehave been wiped out of the visible universe by the subsequent stage of ination. Wewill therefore ignore this stage as practically invisible, and begin the description of the world-as-a-brane universe at curvatures an order of magnitude or so below thefundamental Planck scale. Indeed, at and below these scales, the description basedon the actions ( 7) and (8) should be reliable. The background metric for the (4 +n)-dimensional spacetime which is consistent with the symmetries of the brane-bulksystem is of the form

gµν =1

−a(t)2gIJ

−b(t)2gij

, (9)

where a is the scale factor of the 3-dimensional space, and b is the scale factor of theinternal n-dimensional space, with geometry set by gij where det( gij ) = 1.

As shown in Appendix A, the equations of motion for the coupled a(t), b(t)systemcan be written in the form,

H a + nH a H b + 3 H 2a =1

2(n + 2) bn M (n +2)∗

b∂V ∂b −(n −2)V (10)

andH b + 3 H a H b + nH 2b = −

1(n + 2) bn M (n +2)

∗

bn

∂V ∂b −2V (11)

where we have introduced the Hubble parameters H a ≡ a/a and H b ≡b/b for the two

scale factors, and an overdot denotes a derivative with respect to t. These equationsof motion are supplemented by the constraint equation

6H 2a + 6 nH a H b + n(n −1)H 2b =V

bn M (n +2)∗

. (12)

8



Note that in these formulae the potential V is the effective 4-dimensional potential withmass dimension [V ] = 4 – i.e. , that arising from projecting the bulk energy densitythat results from Lmatter onto our wall.

The natural inationary initial conditions consist of taking the initial (4 + n)-

dimensional universe to be relatively smooth, at and potential energy-dominatedover scales roughly given by the fundamental Planck length 1 /M

∗, and not much

greater [19, 20]. In the brane language the initial conditions we adopt mean thatwe take a portion of the brane of linear size ∼1/M

∗embedded in a (4+ n)-dimensional

volume of the same linear size, to be relatively at, smooth and straight. Indeed, it ismost natural to suppose that the universe starts as a small (4+ n)-dimensional domainof linear extent 1 /M

∗in all directions, and then undergoes an epoch of inationary

expansion almost immediately, at least in the directions parallel to our brane.Quantitatively these conditions can be stated in terms of the initial conditions for

the brane and bulk horizons as follows. The size of initially isotropic and homogeneouscausal domains must be

H −1a,i ∼H −1

b,i ≥M −1∗

(13)

Moreover we can also dene the initial conditions for the scale factor of the brane a andthe radion eld b by referring to the atness problem. Since close to the fundamentalPlanck scale the energy density on the wall and in the bulk is of order unity in funda-mental Planck units, if we pick the gauge such that any intrinsic spatial curvature onthe brane or in the bulk is k = ±1, it cannot exceed in magnitude ρ p, leading to theconditions that ‡

a i∼bi ≥M −1∗

(14)

It is easy to recognize the conditions ( 13) and (14) as precisely the consistent initialconditions for ination, in the case when the fundamental Planck scale is M

∗. How-

ever these initial conditions are merely a rough estimate coming from the requirementthat the approximation based on semiclassical gravity is valid, and that the usual cos-mological problems (horizon, atness, homogeneity etc. .) are solved by subsequentination.

Further, the generation of sufficient density perturbations δρ/ρ ∼δρ/ρ |COBE ∼10−5,

‡In spatially at FRW universes the magnitude of the scale factor a is physically meaningless.However, the spatial curvature, namely the quantity k/a 2 is physical. Thus choosing the value of aat some instant t corresponds to specifying curvature of spatial hypersurfaces. Equivalently, choosingthe value of the constant k, by normalizing it to ±1, corresponds to picking the units for the scalefactors. We will assume this throughout this work, even if we do not explicitly specify the spatialcurvatures (which can be ignored after prolonged ination).

9



implies, at least without the unappetizing introduction of very light elds on our wall,or of the assumption of very unusual initial conditions, that the inationary epoch thatsolves the atness and horizon problems occurs when the size of the internal dimensionsis still relatively small. The basic reason for this is that in a theory with fundamental

Planck scale M ∗, the energy density localized on the brane should not exceed M 4

∗.Similarly the energy density in the bulk should not exceed M n +4∗

. Moreover, the(matter) energy density of the universe at its’ birth is expected to be of order M n +4

∗.

If ination occurs when the internal dimensions are already large, then the effectiveNewton’s constant in our 4-dimensions is already very small GN,eff = 1 / (M n +2

∗bn )

1/M 2∗

, and energy densities of order M 4∗

or less will lead to a very small Hubble constant,and thus typically unacceptably small density perturbations. §

To see this in detail consider the expression for the density perturbations generatedduring slow-roll ination driven by a (canonically normalized) eld ϕ

δρρ

=5

12πH 2in

ϕ. (15)

(The use of this 4-dimensional expression will be self-consistently justied later on inour analysis.) The Hubble constant on our brane during ination is given in order of magnitude by H in V/(M n +2

∗bn

I ), where bI denotes the size of the internal dimensionsduring the inationary epoch. Since V <

∼M 4∗

if bI M ∗

1 then δρ/ρ is very small unlessϕ is extremely small. Although this is a logical possibility, it requires extraordinaryne-tuning and we will not consider this case here. On the other hand if bI M

∗is O(1)

then sufficiently large density perturbations easily result. Moreover, from the powerdependence of H in on bI we see that if the size of the internal dimensions changessignicantly during ination the spectrum of density perturbations will be very far from scale invariant. Since the spectral index of density perturbations nρ is constrained bythe cosmic microwave background (CMB) and large scale structure measurements tobe not signicantly different from the scale-invariant value nρ = 1,

|nρ −1| < 0.2, (16)

it is necessary that the evolution of the internal dimensions is slow compared to that

of our scale factor a: H b H a . A successful phenomenology thus results if the ratioH b/H a approaches a zero for a (small) range of b around the value bI . Let us expand

§Note though that if the internal dimensions are large b 1/M ∗ a small bulk (4 + n)-dimensionalenergy density still allows the effective “projected” energy density on the wall to exceed M 4∗, withoutrequiring the use of the full quantum theory of gravity; the semiclassical approximation is still good.

10



the ratio around this point

H bH a

= S + T bbI −1

2

+ ... (17)

The dimensionless parameters S and T will be bounded in size by the requirementsthat the spectral index n of the density perturbations is close to 1, and that sufficiente-folds of ination occur to solve the horizon and atness problems. Furthermore H bmust be small as compared to H 2a , otherwise the spectral index would again too quicklydeviate from 1.

To quantify these restrictions let us return to the equations of motion for a(t) and b(t)and the general expression for the density perturbations. Successful ination requiresthat the number of efolds N e of superluminal expansion be sufficiently large. If theinitial and nal values of the scale factors just before and just after the inationary

epoch are denoted by subscripts ( i, f ) respectively, then

N e = a f

a i

daa

(18)

= bf

bi

H aH b

dbb

bf

bi

1S + T (b/bI −1)2

dbb

.

To a good approximation ination begins and ends when the condition H b/H a 1 isviolated. Hence to obtain ination, there are two possibilities. Either S must satisfyS 1, which in turn implies

bi,f

bI = 1 T −1/ 2. (19)

The alternative is to have S small (but not excessively) while T must be extremelysmall, which corresponds to slow, power law, ination. Substituting these endpointvalues into ( 19) and performing the integral gives the constraint on S and T arisingfrom N e :

N e1

S + T 2√T tan −1(1/ √S )

√S −log(1 + 1/S ) + 2 log(1 + 1 / √T ) (20)

Requiring, say, N e > 100 then puts an upper bound on the size of ( S, T ). In variouslimits this relation becomes easy to state explicitly. For example if S = T 1 then

N eπ

2T (21)

11



while if S is very small, while T remains O(1),

N eπ

√S −log(1/S ). (22)

In any case obtaining N e

∼

100 efolds of ination requires either S T <

∼

0.02, orT O(1), and S <

∼10−3 . So in practice to get a sufficient number of efolds S should

be somewhere in the range 0 .001 <∼

S <∼

0.02 (or, of course less) depending on T .Now let us turn to the constraint arising from the magnitude and spectral index

of the density perturbations. To do this in detail, we must make some assumptionsabout the identity of the inaton. Since we have argued that ination should occurfar away from the eventual stabilization value of b, the most natural candidate for thepotential energy that drives ination is the radion potential itself, with therefore theradion playing the role of the inaton. The correctly normalized eld ϕ correspondingto the scale factor b is given by (see Appendix A)

ϕ = 2n(n −1)M (n +2) / 2∗

b(n−2) / 2b. (23)

Substituting this into the expression for the density perturbations ( 15) we nd

δρρ

=5

12π 2n(n −1)H 2a

M (n +2) / 2∗

b(n−2) / 2b(24)

512π 2n(n −1)

H aM ∗(M ∗bI )n/ 2S

,

where in the last step we have used ( 17) and the fact that we are close to bI . Requiringb(t) to be essentially static during ination, and in particular that H b/H a and H b/H 2

a

are small near bI is equivalent, using the equations of motion, to the statement that

2 bn ∂ bV −2V

3 (b∂ bV −(n −2)V ) b bI

1. (25)

which translates into the condition that

b

n∂ bV

−2V

bI

V (bI ), bI ∂ bV (bI ). (26)

Therefore ( b∂ bV )|bI 2nV |bI , and up to small corrections, the Hubble parameter H aduring ination is given by

H 2aV (bI )

6bnI M (n +2)∗

. (27)

12



Finally this together with ( 25) gives the expression for the ination generated densityperturbations

δρρ

512π

1S (M

∗bI )n

V (bI )12n(n −1)M 4

∗

1/ 2

. (28)

The spectral index nρ is dened by the comoving wavenumber dependenceδρρ ∼

k(n ρ −1) / 2 (29)

where at horizon crossing we have the relation k/a = H a . From ( 29) we can extract amore convenient expression for nρ:

nρ −1 = 2d log(δρ/ρ )

d log(a)(30)

2S d log(δρ/ρ )

d log(b),

where in the second line the parameterization for H b/H a (17) has been used. Applyingthis formula to the expression for the density perturbations ( 28) leads to

nρ −1 = 2S −n +1

2V dV

d log(b). (31)

But from the two expressions for the ratio H b/H a , one from the equations of motion(10,11), and the other given by the parameterization ( 17) we have, after some algebra,

12V

dV d log(b)

= n −n(n + 2)4

S + T (b/bI −1)2 (32)

and thereforenρ −1 −

n(n + 2)2

S 2 + ST (b/bI −1)2 (33)

This expression together with the experimental constraint |nρ −1| < 0.2 forces S <∼

0.1,thus ruling out the case of T 1, S ∼O(1) allowed by the earlier condition of asufficient number of efolds of ination. Note however that the solutions with T ∼O(1)and S <

∼0.01 automatically satisfy |nρ −1| < 0.2 over essentially the whole range of

a(t) ination. (Experimentally all we require is that |nρ −1| < 0.2 in the range of scales between the COBE and large-scale structure measurements – roughly 10 efoldsrather than the full duration of ination.)

Finally we can use the magnitude of the measured CMB uctuations to constrainthe size bI of the internal dimensions during the inationary epoch. Recall that COBE

13



and the other CMB measurements tell us that at the time the scales k 7H now werebeing inated outside of the horizon, the density perturbations were of size

δρρ COBE

2 ×10−5 . (34)

This together with the formula for δρ/ρ (28), and the bound on the parameter S leadsto a constraint on a combination of the size of the potential during ination V (bI )/M 4

∗and the volume V I = ( bI )n of the internal dimensions, both in units of the fundamentalPlanck scale:

V I M n∗

5 ×104 0.02S

V (bI )M 4∗

1/ 2

. (35)

For S T 0.02 and V (bI ) having a perfectly reasonable value of V (bI ) (200 GeV)4,(for example if M

∗1 TeV) we therefore discover that the epoch of ination gener-

ating COBE needs to occur around the value bI 103/n M −1

∗

, i.e. , when the internaldimensions are still relatively small as expected.

Now let us return to the justication of the use of the usual 4-dimensional expression(15) for the density perturbations. If the de Sitter horizon ( H a )−1 during ination ismuch smaller than the size of the internal dimensions then the full (4+ n)-dimensionalexpression for the density uctuations must be employed, while if ( H a )−1 bI thenit is correct to use the long-distance, effectively 4-dimensional, expression. From theconstraint arising from δρ/ρ (35), and the expression for H a during ination ( 27) wend that

H a bI

(5

×10−3)(n−2) /n

√6S

0.02

(n−2) / 2n V (bI )M 4∗

(n +2) / 4n

. (36)Thus H a bI is always substantially less than 1 for the range of parameters of interest,and the 4-dimensional description of the generation of δρ/ρ is correct.

2.1 Ination from the Einstein frame perspective

Most dynamical aspects the world-as-a-brane scenario are the most transparent interms of the geometric variables employed so far, which can be referred to as thestring-frame quantities. The reason is that the kinematics in this frame is automatically

expressed in terms of the units felt by the observers which live on the wall. In particular,such observers choose to dene scales using their own masses and Compton wavelengthsas yard sticks, and in the string frame these yard sticks are time independent. However,it is illustrative to consider geometrical evolution of the universe in light of the referenceframe where the gravitational sector of the theory coincides with classical Einstein

14



gravity. In this frame, the equations of motion resemble the coupled gravity-mattermodels considered in the context of non-minimal theories of gravity [ 21], and the resultsobtained from them can be a useful addition to physical intuition.

For this reason, here we will recast the description of the inationary stage into the

Einstein conformal frame of the theory on the brane ( 7), (8). This frame is denedby the requirement that the propagator of the graviton is canonical. The map whichcasts the kinetic term of the graviton into the canonical form is

gµν =M ∗

M pl

2

(M ∗b)n gµν (37)

Explicitly, we can dene the Einstein frame comoving time and scale factor accordingto

dt =M ∗

M pl(M ∗b)n/ 2dt a =

M ∗

M pl(M ∗b)n/ 2a (38)

where all barred quantities refer to the Einstein frame. Now, the radion eld is equiv-alent to a scalar eld, dened by

M n +2∗

bn = M 2pl exp − n2(n −1)

ϕ

M pl(39)

Note that the normalization employed here is dictated by the denition of the pertur-bation of the radion away from its mean value during ination, which sets the effectivebackground Planck mass. We note that the initial conditions for ination in the Ein-stein frame, in the units of the effective Planck scale, require homogeneity and atnessover distances of

∼

lPl,eff = 1 /M pl , which can be recognized as a usual inationaryinitial condition.

This Einstein frame picture is very useful to compute the density contrast in modelswhere radion is the inaton. The density contrast is given by the standard formula

δρρ

=H 2

2πϕ= 8

H 2

ϕ(40)

in our normalizations, and where the prime is the derivative with respect to t. Todetermine the density contrast in the string frame, we recall that it is, roughly, con-formally invariant during ination [ 22], and conformally transform ( 40) to the string

frame. The radion and the Hubble parameter transform to

ϕ = − 2n(n −1)M 2pl H b

M ∗(M ∗b)n/ 2

H =M pl

M ∗(M ∗b)n/ 2 H a +

n2

H b (41)

15



which is straightforward to determine from their denitions. With this, we nd

δρρ

=8

2n(n −1)

(H a + n2 H b)2

M ∗(M ∗b)n/ 2H b

(42)

Now we consider the tilt of the perturbation spectrum. To consider it, we dene theeffective slope by

θ∼

δρρ (ta )δρρ (tb) ∼

b(tb)b(ta)

n/ 2

(43)

where the last equality arises because of the slow roll conditions H a,b (ta) ∼H a,b (tb)during ination. Since b(ta)∼b(tb)(1+ H bδt) and since we compare the tilt between 50and 60 efoldings, we get δt∼10H −1

a . Thus roughly, θ∼1 −5n H bH a On the other hand,

during this time θ changes according to θ∼kakb

(n ρ −1) / 2, and given that kk∼akH a , we

nd that the ratio of the wave vectors is given by ka /k b

∼

aa /a b

∼

exp(H a δt)

∼

exp(10).Taking this and the bounds on the spectral index nρ, nρ ≤1 ±0.2, we nd that

θ∼e±1/ 2. (44)

This leads to the following inequality which the expansion rates must satisfy duringination:

H a >∼

5n1 −exp(±1/ 2)

H b ≥15nH b, (45)

As a result, the density contrast δρ/ρ is

δρρ 8

2n(n −1)H

2a

M ∗(M ∗b)n/ 2H b

(46)

The COBE data tell us that δρ/ρ ∼10−5 . Using this and H a ≥15nH b, we nd

H a <∼

10−7M ∗(M ∗b)n/ 2 (47)

We can use this inequality to obtain a bound on the size of the internal dimensions bI

during ination. We repeat that the evolution is similar to the Einstein frame thanksto the slow roll conditions. Hence the vacuum energy density during ination is

ρvacuum ∼M (2+ n )∗

bnI H 2a (48)

Using our estimate for H a ,

ρvacuum ∼10−14 M 4∗(M ∗bI )2n (49)

16



However ρvacuum must not exceed M 4∗. In fact, since initially the theory is very close to

the quantum gravity scale, in order to insure the validity of the semiclassical approxi-mation the energy density cannot exceed, let’s say, 10 −5M 4

∗, meaning that the energy

scale is about a factor of 105/ 4 below the quantum gravity scale. This ensures that the

semiclassical description is correct. Using the equation for the energy density in termsof bI above, we obtain the inequality

bI <∼

109/ 2n M −1∗

(50)

Note that the upper limit on bI ranges between about 200 M −1∗

for n = 2 to about6M −1∗

for n = 6. In either case however, these are clearly the correct initial conditionsfor ination, which in fact come naturally in this context.

3 Post-ination evolution to stabilization point

As we have discussed in the previous section, when the inationary stage ends, the ratioof internal space to on-brane Hubble rates of expansion H b/H a approach unity. Fromthis time on, the slow roll conditions for the effective potential cannot be upheld anymore. The kinetic energy stored in the expansion of the internal space begins to playa signicant role in the evolution of the brane-world. Rather interestingly, however,we will see that under rather general conditions the expansion of the whole higher-dimensional universe under the combined inuence of the potential and kinetic energiesduring this era is well approximated by a generalization of the well-known Kasnersolutions. These solutions generically describe cosmic evolution which is anisotropicin different directions, with a subset of the directions contracting while the othersexpand. Specically we will nd that after the end of the initial brane-world inationthe directions longitudinal to the brane contract, while the internal directions expand,both according to some power-law time dependence. The precise form of the powerlaws is controlled by the codimension of the brane ( n in our notation) and the leading-order behavior of the stabilizing potential V (b) as a function of the scale factor b of the internal dimensions.

3.1 Theory of the era of contraction

To illustrate generic features of such behavior, and show why it should be immediatelyexpected, at least in a subset of cases, consider the limiting case where after the exitfrom the inationary stage the potential V (b) drops by many orders of magnitude.

17



The evolution of the scale factors a(t) and b(t) is then controlled entirely by the kineticenergy, and the equations of motion appropriate for this case are just the higher-dimensional vacuum Einstein equations,

Rµν = 0 (51)

In other words ( 10,11) with V set to zero. The (4+ n)-dimensional solutions which areconsistent with the brane symmetries are of the form

ds2 = −dt2 + a2i

tt i

2k

dx23 + b2

itt i

2

dy2n (52)

where a i , bi are now the “initial” values of the scale factors as set by the end of the ina-tionary stage. The powers k and are uniquely determined by the Einstein equations,which reduce to two simple algebraic equations,

3k + n = 1

3k2 + n 2 = 1 (53)

A metric of the form (52) with exponents satisfying ( 53) is known as the Kasnersolution. The solutions of the algebraic equations for the exponents are

k =3 3n(n + 2)

3(n + 3)

=n

± 3n(n + 2)

n(n + 3) (54)

Phenomenologically, we certainly need the internal dimensions to grow in size andapproach the stabilizing value, and this selects the upper sign in the equalities ( 54),which then implies that our longitudinal brane directions contract. As we will soonsee when we re-introduce the potential V (b), this behavior is physically selected bythe asymmetry embodied in the potential, and in particular the fact that it must havea minimum at the stabilizing value b0, with bn

0 = M 2pl /M n +2∗

. In any case in thissimple potential-free Kasner case, the explicit values for the powers range between

k = −0.1266, = 0 .69 for n = 2, and k = −1/ 3, = 1 / 3 for n = 6.It is useful to express the brane scale factor a(t) as a function of the size of the

internal dimensions b(t),aa i

=bbi

−|k|/(55)

18



The amount of contraction of our scale factor is thus determined by the increase in b(which in turn is determined once the end of ination value of b = bi is specied), andthe known power −|k|/ . Note that for n = 2, k/ ∼−0.183, and so the brane worldcontracts by only one order of magnitude for almost six orders of magnitude of radion

increase. As n increases, this dependence speeds up, and for n = 6 the contraction of the brane dimensions and expansion of the internal dimensions are essentially equal inmagnitude (we have of course ignored small transient effects at both the start and endof this Kasner phase which slightly modify the above relationships).

We are typically interested, however, in the case where there exists a non-trivial V (b)potential as well as matter on our brane. We again take the metric to be of the form

ds2 = −dt2 + a2(t)dx23 + b2(t)dy2

n (56)

and assume that the matter on the wall is represented by a classically conserved, perfect

uid energy-momentum tensor T µν = ( ρ + p)uµuν + pgµν , µT µν = 0, where uµ is afuture-oriented timelike vector, with the components uµ = (1 , 03+ n ) in the basis ( 56).Here ρ is the energy density of the wall matter and p = γρ is the pressure, with γ aconstant given by the speed of sound on the wall. The equations of motion slightlygeneralize (10) and (11),

6H 2a + n(n −1)H 2b + 6 nH a H b =V + ρ

M n +2∗

bn

bb

+ ( n −1)H 2b + 3 H bH a =1

M n +2

∗

bn2V

n + 2 −b

n(n + 2)∂V ∂b

+ρ −3 p

2(n + 2)aa

+ 2 H 2a + nH bH a =1

M n +2∗

bnb

2(n + 2)∂V ∂b −

n −22(n + 2)

V +ρ + ( n −1) p

2(n + 2)ρ + 3 H a ( p + ρ) = 0 . (57)

The last equation is of course the usual wall energy-momentum conservation equation,µT µν = 0.It is important to note that the contraction of the brane in Kasner-like solutions

leads to an increase of the energy density of any matter or radiation present on thebrane at the beginning of the contraction, and this will lead to “bounce” solutions

that will be important to us later. Also note that these equations generically receivequantum corrections from particle production via curved space effects or the conformalanomaly. The corrections would manifest themselves as a nonzero source term on theRHS, which as we will see would lead to signicant effects only early on, and thereforecould be modeled by an appropriate choice of the initial condition for ρ.

19



Namely, the energy density on the wall corresponds to the particles produced bychanging gravitational elds at the end of ination. Since these phenomena are es-sentially similar to Hawking radiation, the value of ρ as compared to the potential issuppressed by a factor of H 2a /M n +2 bn at the end of ination. Using the COBE data to

constrain this quantity, we see that initially ρ is much smaller than V . Hence ignoringit is an excellent approximation. For most of this stage, therefore, the dynamics of the universe is determined by the interplay of the radion kinetic and potential energy,which completely determine the evolution of the wall geometry. We will simply carryon with the analysis of ( 57), treating ρ as a small perturbation and ignoring its back-reaction on a and b. However, although to the lowest order, the particle productionphenomena are negligible, they could lead to interesting effects for reheating. We willreturn to this later.

However before we consider such blue-shifting of the brane-localized energy density,

let us turn our attention to the more general and appropriate case of non-negligibleradion potential. Remarkably in this case a form of Kasner-like behavior still applies.

Quite generically, in the semiclassical limit, fully valid at this stage of the evolution,the potential may be viewed as an expansion in inverse powers of the radion eld b. Atgeneric values of b away from the stabilization point the potential will be dominated bya single term in this expansion. Hence, we can simply approximate V (b) by a monomialof the form V = W b− p, where W is a dimensionful parameter with [ W ] = 4 − p.¶

If we substitute V = W b− p in the equations of motion, ignoring ρ and p for themoment we nd

6H 2a + n(n −1)H 2b + 6 nH a H b = W M n +2∗

bn + p

bb

+ ( n −1)H 2b + 3 H bH a =(2n + p)W

n(n + 2) M n +2∗

bn + p

aa

+ 2 H 2a + nH bH a = −(n + p−2)W

2(n + 2) M n +2∗

bn + p . (58)

These equations can in fact be solved exactly! With appropriate eld redenitionsand gauge (coordinate) transformations, they can be mapped to a system of equationsdescribing the motion of two particles in one dimension, one free, and another Liouvillewith an exponential potential.

First note that in a certain parameter range these exact solutions asymptotically¶ W could well depend logarithmically on b. This mild additional b dependence will not change

either our qualitative, or to a good approximation our quantitative conclusions. For simplicity weignore it in the following discussion.

20



converge to the “potential-free” Kasner solutions ( 52) and ( 54) above, as we discuss inAppendix B. This parameter range turns out to be the one in which, upon substitutionof the power dependence ( 54) into the equations ( 58), the potential terms on the RHSvanish more quickly as a function of time than the LHS. Since the LHS always scales

as∼t−2

this is the case when ( n + p) > 2, or equivalently(n + p)

n(n + 3)(n + 3n(n + 2) > 2 (59)

This gives a curve in (n, p) space above which ( i.e. , for larger values of p) the exactsolution asymptotes to the potential-free Kasner exponents. The critical values of pvary from 0.899 at n = 2 to p = 0 at n = 6.

We now discuss the changes of variables which allow the exact solution of the equa-tions with potential. First dene

a = a ieα (t )

b = bieβ (t )

(60)where a i and bi are as before the initial values of the scale factors at the beginning of the epoch of radion coasting. This means that the appropriate initial conditions forthe dimensionless variables α and β are α i = β i = 0. Further, dening the parameterω = W

M n +2∗ bn + p

i, substituting this and ( 60) into (58), and going to the new time variable

τ :dτ = −e−3α −nβ dt (61)

leads after some simple algebra to the equations of motion in their nal form, suitable

for explicit analysis:6α 2 + n(n −1)β 2 + 6 nα β = ωe6α +( n− p)β

β =(2n + p)ωn(n + 2)

e6α +( n− p)β

α = −(n + p −2)ω

2(n + 2)e6α +( n− p)β . (62)

Here primes denote derivatives with respect to τ . The specic form of τ = τ (t) can bedetermined after the solutions are found.

The equations ( 62) are immediately integrable. Indeed, consider the linear combi-nation 4α + 2n (n + p−2)

2n + p β . By using the second order differential equations, it is easy toverify that 4 α + 2n (n + p−2)

2n + p β = 0. Hence we can immediately write one rst integralof (62):

4α +2n(n + p −2)

2n + pβ = C 1 + C 2τ (63)

21



where C 1 and C 2 are integration constants to be determined later.To nd the other integral of motion, we can dene the new variable

X = 6α + ( n − p)β (64)

The remaining independent second order differential equation becomes

X =6n −4np −n2 − p2

n(n + 2)ωeX (65)

This is the Liouville equation, corresponding to a particle in 1 dimension moving inan exponential potential. This has an easily determined rst integral, simply given bythe conservation of energy:

X 2 = 2∆

n(n + 2)ωeX + E 0 (66)

with E 0 the energy integral, and where we have dened ∆ = 6 n −4np−n2 − p2. At thismoment, we need to make three observations. First, X is a good independent variableonly as long as ∆ is nonzero. If it is zero, the variable X degenerates to the previousrst integral ( 63), up to an overall constant, and so another independent integral shouldbe used. This is easy to take into account however, and besides corresponds to a set of measure zero in the phase space of solutions, and thus we will not pay it much attentionhere. We will instead focus on the more generic cases where X is independent from(63).

Second, the curve in the ( n, p) plane where ∆ = 0 vanishes is precisely the curvedened by (59) that separates the traditional potential-free Kasner solutions from themore general (but still asymptotic power-law) behavior that we discuss below. Inparticular, the traditional potential-free Kasner solutions apply asymptotically in theregion ∆ < 0. The full behavior in this region, including the transient regime beforethe power-law dependence on t sets in can easily be discussed by a simple generalizationof the analysis described below and in Appendix B for the case ∆ > 0. Leaving theregion ∆ < 0 to the Appendix, we focus on the novel case of ∆ > 0 in the following.

Third, the integrals of motion C 2 and E 0 are not independent by virtue of the

Einstein constraint equation, which is the rst equation of ( 62). If we take (66) andthe rst derivative of ( 63), and substitute them into the constraint, we nd

E 0 =3(2n + p)2

4n(n + 2)C 22 . (67)

22



Thus we see that the constant C 2 completely controls the dynamics. With this, wehave essentially reduced the system ( 62) to a functional constraint ( 63) and a simple1st order equation

X 2 = 2∆

n(n + 2)ωeX +

3(2n + p)2

4n(n + 2)C 22 (68)

which can be easily integrated when ∆ = 0.As shown in Appendix B when ∆ > 0, the exact solutions of these equations yield

the long-time behavior of the scale factors a(t) and b(t). It is given by

a = a itt i

−n ( n + p − 2)

( n + p )(2 n + p )

b = bitt i

2n + p

. (69)

These solutions describe a situation in which the on-brane scale factor shrinks whilethe size of the internal dimensions continues to grow. The exact solutions show thatfor C 2 = 0 they are generally valid, while they are asymptotic long-time attractorsfor the generic cases with C 2 < 0 (i.e. , the case in which both H a and H b > 0 afterthe end of ination). This means that even if the brane scales initially continue toexpand, soon after the end of ination they inevitably start to contract. Note thatthe t-dependence of b(t) is exactly such as to have the potential-dependent RHS of theevolution equations scale in the same 1 /t 2 fashion as the LHS.

In any case, the main point of this analysis is that the evolution after the initial stageof ination continues into a phase of slow progress of b(t) towards the stabilizationpoint, with, under quite general conditions, a simultaneous contraction of our branescale factor a(t).

The amount of a(t) contraction is controlled by the exponents in ( 69) (or in thecase of ∆ < 0 the exponents in ( 54)), and the amount of expansion of b until thestabilization point is reached. We can use the asymptotic form of the solutions toplace an upper bound on the amount of contraction of the brane as a function of theevolution of b. We have

aa i ≤

bi

b

ζ

(70)

where the parameter ζ is given by

ζ =n(n + p−2)

2(2n + p)for ∆ > 0

If ∆ = 0 a simple modication of this procedure still yields the second integral of motion.

23



ζ =3n − 3n(n + 2)

6for ∆ < 0. (71)

In these equations recall that ∆ = 6 n −4np −n2 − p2, while the effective 4-dimensionalpotential for b varies as V

∼

b− p. It is easy to see that the exponents for the a(t)

and b(t) evolution calculated in the ∆ > 0 and ∆ < 0 regions are continuous on thecurve ∆ = 0, and thus so is ζ . Furthermore, note that for a given n the greatest a(t)contraction occurs in the ∆ < 0 case.

3.2 Phenomenology of the era of contraction

There are a number of interesting consequences of the period of a(t) contraction justdescribed. First, during the contraction era, the brane universe looses a number of efoldings of the scale factor to the shrinkage. This implies that the early period of

ination needs to produce that many efoldings more than the naive minimum neededto solve the horizon and atness problems. Moreover, the contraction of a(t) blue-shifts any energy density left on our wall at the end of ination, and as we will arguebelow, this could, together with the remnant of the de Sitter era Hawking radiation,conceivably be the source of reheating of our brane.

To quantify these remarks, we note that the contraction factor of the brane cannotexceed the amount

a f

a i ≤bi

b0

ζ

(72)

where the fact that the nal value of b(t) is well-approximated by the stabilizing value,b0, has been used. The value of bi is set by the size of the internal dimensions at the endof the early period of ination and at worst is of order the fundamental Planck lengthbi ≥M −1

∗. (The COBE constraint ( 35) typically requires a larger value of bi , and this

just lessens the amount of contraction.) Thus, the maximum amount of contraction isbounded from above by

a f

a i ≤M ∗

M pl

2ζ/n

. (73)

The numerical value of this formula ranges between about 900 for n = 2 to about

2 ×105 for n = 6. This means that we loose to contraction at most about 7 efoldingsof ination for n = 2 up to about 12 for n = 6. Clearly, this is not excessive, and canbe easily made up for by a slightly longer period of early ination.

We now turn to the evolution of the wall and bulk energy densities during the periodof contraction. First note that the dominant form of energy density at the end of the

24



inationary period is the bulk energy of the radion, i.e. the b(t) scale factor kinetic andpotential energy. The (effective 4-dimensional) kinetic energy density of the radion is

ρb,KE = n(n −1)M n +2∗

bn (H b)2 (74)

while its 4-dimensional potential energy density is just V (b) as used in the previoussections. ¿From the exact solutions presented in the previous subsection we nd thatscaling of the energy density of the radion depends on the exponents ( k, ) for theevolution of the scale factors a∼tk , and b∼t , which in turn depend on whether thepotential is important or not, in other words the sign of ∆. The result is

∆ ≥0 (potential case) (75)

ρb,PE ∼b− p, ρb,KE ∼b− p

∆ < 0 (simple Kasner)

ρb,PE ∼0, ρb,KE

∼b2n−√3n (n +2) .

So far we haven’t discussed what ends the contraction period. As shown in detail inAppendix C, it is a remarkable fact that contraction of a(t) stops and reverses whenρwall satises∗∗

ρb = n(n −1)M n +2∗

bn H 2b = ρwall . (76)

There are two generic possibilities for how this condition may come to be satised:The rst takes the primordial ρwall left over from the inationary epoch. If there issufficient a(t) contraction, this inationary ρwall can becomes comparable to ρb before breaches the stabilization point, b0. Then a “Big Bounce” occurs, the contraction stops,and a modied expansionary phase that we discuss below begins. This expansionnally becomes the usual FRW expansion after b reaches b0. The second possibility isthat a form of reheating takes place on the wall which is totally unconnected with thecontraction of a(t), but that again leads to ρwall ≥ρb. Possibilities in this class includethe decay of some metastable state on the wall, or the collision of some other branewith our brane.

Consider the rst, model-independent, scenario. We are used to thinking of theuniverse at the end of ination as being very cold, but due to the de Sitter era Hawkingradiation this is not completely so. What is the initial value of the ratio ρwall /ρ b soproduced? At the end of ination, our brane is not empty but inevitably contains∗∗The exact analysis we have performed proves that this is the case when the dominant form of

radion energy is kinetic, and, moreover, as we will show in Section 3.4, necessarily and automaticallyoccurs if the radion stabilizes.

25



a radiation bath left over from the early inationary de Sitter phase, whose energydensity at the end of the de Sitter phase has been estimated by Ford [ 23] (see also[24]), with the result

ρwall,i 10−2(H a,I )4, (77)

depending on the exact nature (conformal or not) of the coupling between the matterand metric. Here H a,I is the Hubble parameter at the exit from ination. On the otherhand, the initial value of the radion energy is of order

ρb,i n(n −1)M n +2∗

(bI )n (H b,I )2 6M n +2∗

(bI )n (H a,I )2, (78)

since ination terminates when H b/H a approaches unity. Furthermore due to the slowroll conditions, at ination exit H a is roughly the same as during the inationaryphase. Given these expressions for the initial energy densities, equations ( 27) and(28) together with the constraint of reproducing the CMB/COBE data lead to the

interesting relationshipρwall

ρb i

n(n −1)6

S 2δρρ COBE

2

. (79)

Due to the contraction of our scale factor, the wall-localized radiation can becomeimportant, the ratio ρwall /ρ b approaching unity. Using the fact that ρwall ∼a−4, andthe relation ( 70) between the evolution of a and b, we nd that it scales as

ρwallρb ∼

b p

a4 ∼b(2 n 2 +4 np + p2 −4n )/ (2 n + p) for ∆ > 0

ρwallρb

∼

b−2n + √3n (n +2)

a4∼b

√n (2+ n )/ 3

for ∆ < 0. (80)Given the initial value ( 79) of the ratio we can compute with the aid of the scaling

laws (80) how it behaves as the contraction proceeds. Putting in the experimentallyobserved value of δρ/ρ ∼2 ×10−5 , and taking a reasonable value of S ∼0.01, we ndthat in the simple Kasner ∆ < 0 case, ρwall = ρb before b b0 for all n = 2 , . . . , 6. Onthe other hand, in the case of the Kasner-like solutions with potential, ∆ > 0, the rateof increase of the ratio ρwall /ρ b is always slower as a function of b, and in some cases(e.g. , n = 3, p = 0) the stabilization point is reached before ρwall ∼ρb. We will focuson the ∆ < 0 cases for illustration in the following.

Specically, using the scaling laws for the evolution of the energy densities, ρwall = ρb

at a value of b = bf given by

bf

bi= S

δρρ COBE

−2√n (n +2) / 3

. (81)

26



We can now compute the energy density on the wall at the end of contraction:

ρwall,f 7 ×102M 4∗(bI M

∗)2n n(n −1) S

δρρ COBE

(2( n +1) 2−4n√n (n +2) / 3)

. (82)

In terms of the temperature of radiation at the end of the contraction phase, thisbecomes

T wall,f 6M ∗(bI M

∗)n/ 2(n(n −1))1/ 4 S

δρρ COBE

(( n +1) 2 / 2−n√n (n +2) / 3)

. (83)

If we use the fact that the typical volume (measured in units of the fundamental Planckmass M

∗) of the internal dimensions during ination is at most ( bI M

∗)n 103 , and

that Sδρ/ρ < 2 ×10−7, given COBE and S <∼

0.01, we nd that the wall temperatureat the end of contraction is bounded above by 1 .4×10−6 , 7×10−7 , 4.5×10−6 , 3×10−4,

and 0.22 in units of M ∗, for n = 2 ,..., 6 respectively.We compare this to the limits found in [ 3], which showed that the temperature of

radiation on the brane, by stabilization , cannot exceed a certain maximal value in orderto prevent overproduction of bulk gravitons by evaporation. This temperature, calledthe normalcy temperature T

∗in [3], was dened as the temperature above which the

cooling of the wall by graviton production begins to compete with the normal adiabaticcooling by expansion. It was estimated to range between about 10 MeV for n = 2 toabout 1GeV for n = 6. For M

∗∼1 TeV, the above expression for the reheatingtemperature of the wall radiation at the end of contraction gives about 1 .4MeV for

n = 2 up to about 300MeV for n = 5 and 220GeV for n = 6. In fact, if we takeM ∗

30TeV for the n = 2 case, as suggested in [3], we can see that the upper boundon the predicted wall reheating temperature becomes 45MeV. Thus we see thatapart from the n = 6 case we certainly satisfy the normalcy bounds on the reheatingtemperature.

However, these numbers are only the upper bounds for a number of reasons. First,in reality, the evolution is not always Kasner-like. Instead, there are periods immedi-ately after the end of ination when the brane scale factor is increasing, or decreasingmore slowly than in the Kasner case. As a result of this, the actual blue-shift is less

than the bounds used above, and this can easily lower the upper bound by an O(1)factor. Second, after the end of contraction, and until the stabilization point b = b0

is obtained, both a(t) and b(t) expand. Since the normalcy bounds strictly only applyafter stabilization is reached, and the expansion of a(t) red-shifts the temperature, theabove estimates of T wall are actually too high.

27



Having found reassurance in these numbers, we can ask if this reheating could benot only indifferent, but benecial for late cosmology. Namely, the bounds we havederived above suggest that the reheating of the universe on the wall due to contractionmay be just sufficient to warm the world enough so that nucleosynthesis can occur

without hindrance. Indeed, the upper bound on the reheat temperature is alwaysabove the nucleosynthesis scale, suggesting a very interesting possibility that in thispicture reheating and particle production could be purely gravitational. In conventionalcosmological models based on quantum gravity at the scale 10 19 GeV, gravitationalparticle production is typically insufficient for reheating, largely due to the fact thatination occurs early, and that between it and the nucleosynthesis era the universe hasexpanded by many orders of magnitude, diluting the particles produced in the earlyde Sitter phase. In our model, in contrast, the part of the world which contains ouruniverse contracts, increasing the energy density and number density on the wall rather

than diluting them.

3.3 Bulk graviton production

We close this section with the consideration of production rates for bulk gravitonsduring the era of radion evolution. There are actually two slightly different issues herethat we have not so far distinguished: i) the energy density in the (would-be) zero modeof the bulk graviton, namely the radion, and, ii) the energy density in Kaluza-Klein(KK) excitations of the graviton in the bulk.

The constraints on the energy density in KK excitations is actually more severethan that on the radion energy density, which just comes from overclosure. The reasonfor this is that even though the lifetime of the bulk KK modes is very long, a smallfraction of them will decay back to photons on our brane, causing distortions in thediffuse gamma-ray background [ 3]. The diffuse gamma-ray background constraint onlyapplies to the excited KK modes, since in general it is only the excited states that candecay to dangerous energetic SM states on our brane. (Recall that typically the massof the radion, the would-be zero mode, is >

∼10−3 eV.) This constraint is more severe

than the overclosure constraint which applies to both the radion zero mode and all theKK excitations together.

In any case in the next section we will consider the overclosure constraint on thetotal bulk energy, and see that generically there is a problem. Here we want to checkthat the bound on the KK excitations of the graviton is automatically satised. (Theanalysis will show in passing that the dominant form of bulk energy will be in the

28



radion motion, and not the KK excitations.)First consider the evolution of the projected energy density (of mass dimension

[ρKK ] = 4 as usual) of the bulk KK gravitons in the absence of particle production:

ρKK + 3 H a ρKK + H bρKK = 0 . (84)

The second term on the LHS corresponds to the usual 1 /a 3 dilution of massive particles(which from this 4-dimensional perspective the KK excitations of the graviton appearto be; see [3] for a full discussion). The novel feature of this equation is the third termwhich expresses the fact that as the internal radii increase, the KK masses decrease as1/b . Another way of saying this is that the KK mass is really the quantized momentumin the internal directions, and this red-shifts as b(t) expands.

As recognized long-ago by Parker and Zeldovich, and claried later by many au-thors [24], a time-dependent gravitational eld can produce particles from the vacuum,by essentially a version of the pair-creation process that takes place in strong electricelds for charged particles. Up to order one coefficients that will not be important, themagnitude of this particle production per unit time per unit volume is given by H 5,where H is the typical Hubble constant. †† Thus the equation for the projected KKenergy density becomes

ρKK + 3 H a ρKK + H bρKK = H 5. (85)

It is easy to solve this equation. Using the substitution ρKK = f a ia

3 bib , it can be

reduced to ˙f = H 5a i

a

3 bi

b(86)

which, employing the Kasner era power laws, and noting that the resulting power of tis less than −1, gives upon integration a result dominated by the early stages

f ∼1t4

i −1t4 H 4i (87)

for all power-law solutions. Converted back into ρKK , this says that the nal energydensity of the KK gravitons in the bulk which came from particle production is bounded

from above by

ρKK,f <∼

H 4ia i

a f

3 bi

bf . (88)

††More precisely, as shown in [ 24], in the anisotropic case considered here, it is the Hubble constantof the contracting dimensions.

29



This differs from our estimate of the nal energy density of the blue-shifted wall-localized radiation at the end of the epoch of contraction, only in that it is furthersuppressed by a factor of ( a f /a i)(bi /b f ), which comes from the fact that the KK gravi-tons are red-shifted by the bulk expansion, but only diluted and not blue-shifted by

the wall contraction. Thus we getρKK,f

ρwall,f ≤a f

a i

bi

bf (89)

<∼

101−30/n 1+ ζ ,

where in the second line we have used (70), and the conservative estimates bi∼10M −1∗

and bf = b0. Evaluating this in, for example, the case of the simple Kasner contractionwith exponents given in ( 71) leads to ρKK /ρ wall varying between 3 ×10−17 for n = 2to 1

×10−8 for n = 6. This shows that the effective temperature of the KK gravitons

is well below the diffuse gamma-ray bound, even before any dilution necessary to solvethe radion moduli problem. It also demonstrates that the vast majority of the energyin the bulk is in the motion of the zero mode radion ρb, rather than in the bulk KKmodes. This is simply because we have shown above that ρb ρwall is the naturalcircumstance at the end of the epoch of contraction.

3.4 The era of contraction and stabilization in the Einsteinframe

The epoch where the radius grows from its initial small size to its nal value can alsobe simply and physically understood in the Einstein frame. (Einstein frame quantitieswill be denoted by an overbar in the following.) Recall that the metric in the Einsteinframe is related to the one in the string frame via

gµν = enβ gµν , b = eβ (90)

(throughout this subsection we work in units with M ∗

= 1). Using this map, it isstraightforward to relate the quantities in the string frame to their counterparts in theEinstein frame. For instance, it is trivial to check that

H a = en2 β H a −

n2

β (91)

Therefore, we can see that even though in the Einstein frame H a > 0, this can looklike “contraction” in the string frame, i.e. H a < 0, provided that β is large enough. On

30



the other hand, as long as the radius eventually stabilizes at its nal size so β = 0, wewill have H a > 0. This proves that, in the string frame, there is always a “big bounce”as long as the radius eventually stabilizes.

Moving on to the dynamics, the action in the Einstein frame is

S = d4y√−g −R+n(n + 2)

2(∂β )2 −e−2nβ V (β ) (92)

We can parametrize the potential as

V = b− pW ≡e− pβ f (β ) (93)

We wish to regard f as very slowly varying and will treat it as a constant duringthe radion “coasting”. Of course, at the nal value of the radion β

∗we must have

f (β ) = f (β ) = 0 , f (β ) > 0.

If we now work with the canonically normalized eld

β = n(n + 2) β (94)

the action becomes

S = d4y√−g −R+12

(∂β )2 −e−xβ f (β ) , x ≡2n + p

n(n + 2)(95)

The equations of motion are now just the familiar ones for a FRW universe with a

scalar eld β with an effective potential V ef f (β ) = e−xβ

f (β ). Writing the scale factoras usual a = eα , we have

6 α2 =12

β 2 + fe −xβ (96)

β + 3 α β = xfe −xβ

Suppose we ignore f ; then β only has kinetic energy and we trivially nd the solutions

α =13

log(t)

β = 2√3log(t) (97)

Obviously, for sufficiently large x, the effective potential V eff is falling off so rapidlythat ignoring f should be a good approximation. We can see what the lower boundon x is by substituting the above potential-free solutions into the exact equations of

31



motion; the kinetic energy terms scale like t−2, whereas the potential energy term scaleslike t−2x/ √3. Therefore, ignoring the potential is a good approximation when

x > √3 →x2 −3 = −∆ > 0 (98)

which agrees with the string frame result. Notice also that this case corresponds to thestring frame contraction, since

sgn(H a ) = sgn α −n

2 n(n + 2)β = sgn

13 −

n

3n(n + 2)< 0 (99)

What about x < √3 or ∆ > 0? In this case, the effective potential does not fallsteeply enough for the energy density to become kinetic energy dominated. What hap-pens instead is that the radion rst gets accelerated by the potential till the kinetic

energy briey dominates, whereupon it gets diluted again and the cycle repeats. There-fore, on average we expect the kinetic and potential energies to be the same. It is easyto see that this assumption is self-consistently justied by the equations. Therefore,we have

β = e−x2 β →β =

2x

log(t) (100)

Inserting this ansatz back into the equations, we nd trivially

α =1x2 log(t) (101)

These solutions correspond to the new, modied Kasner solutions found in the stringframe analysis. Once again, it is easy to see that these solutions correspond to con-traction in the string frame:

sgn(H a ) = sgn1x2 −

n

n(n + 2) x= sgn(2 −n − p) (102)

while from our solutions in the string frame a contracts when n + p−2 > 0.We can now discuss what happens as the radion nears its nal minimum at β = β

∗.

The effective potential V eff can be approximated as quadratic around this nal point

for (β −β ∗)/β ∗<∼1. If the radion approaches this region with kinetic energy sufficientlysmaller than, or comparable to, the potential energy, then by equipartition it will betrapped in the well and will oscillate about the minimum. On the other hand, if thekinetic energy is much larger than the potential at the top of the well, β will escapefrom the region close to the minimum; its subsequent fate depends on the form of V eff

32



at larger values of β . It can either escape to innity, or it may turn back around atsome large distance away and “slosh” back and forth with very large amplitude aboutβ ∗. It is therefore clear that for x < √3, where the kinetic and potential energy stay

comparable throughout the coasting period, the radion will not overshoot. On the other

hand, it appears that for x > √3, where the kinetic energy dominates, the radion couldsignicantly overshoot the minimum. This statement has to be qualied, however, sinceall of the above analysis neglects the effect of the radiation energy density left overafter ination. Of course, in the Einstein frame,this energy does not appear to be blue-shifted, because the Einstein frame scale factor never decreases. However, during thekinetic energy dominated era, the kinetic energy redshifts as a−6 whereas the radiationenergy density only redshifts like a−4, so the radiation will eventually dominate. Letus consider what happens after radiation domination, ignoring the radion potential.Since a∼

√t during radiation domination, the radion kinetic energy β 2∼a−6∼t−3, so

˙β ∼t−

3/ 2

giving β ∼t−1/ 2

. In other words, ignoring the radion potential, the radiationprovides enough friction to stop β . Therefore, if radiation domination happens beforeβ approaches the minimum β

∗, radiation domination prevents overshoot. In all cases

of interest to us this indeed happens; as demonstrated numerically in subsection 3.2,radiation domination takes place for values of the radius b less than the minimum b0.

3.5 Stabilization and the moduli problem

In the previous subsections we discussed a rather interesting mechanism by which theremnant Hawking radiation left over from the de Sitter era could be the source of reheating on our brane when combined with the fact that a(t) goes through a periodof contraction.

However as we noted above, before this mechanism can be viewed as a realistic way toreheat our brane, we must consider the radion eld. At the stage where the contractionstops and the brane undergoes a Big Bounce, the contribution of the energy densityof the radion eld is still a signicant contribution to the total (indeed O(1)). Thisfact leads to a radion moduli problem. The difficulty is that the radion is so lightand weakly coupled that it lives typically much longer than the age of the universe.Since its coherent oscillations about the minimum redshift away only as 1 /a 3, it caneventually overclose universe. In order to avoid this, the energy stored in the radionmust be small, relative to the radiation energy T 4

∗, when the universe is reheated to T

∗:

ρrad∗T 4∗

=ρrad∗T 3∗

×1T ∗

=ρrad 0

T 30 ×1T ∗

<3 ×10−9 GeV

T ∗

(103)

33



where we have used ρ/T 3 = const . and ρcrit 0/T 30 ∼3 ×10−9 GeV. Given that T

∗is

bounded between ∼1 MeV - 1 GeV by normalcy constraints, the energy density inthe radion must somehow be diluted by ∼10−7 −10−9 in order to avoid overclosure.Fortunately this implies that, for example, only about 5 or 6 efolds of late ination

are needed so solve this problem. In any case this moduli problem is one of late (post-stabilization) cosmology, rather than early pre-stabilization cosmology; we will returnto address it in a future publication.

4 Conclusions

We have argued that early ination when the internal dimensions are still small cansuccessfully accomplish all that is required of ination, including generation of suitableδρ/ρ without the unpleasant introduction of very light or ne-tuned wall elds. Indeed,

the very fact that the internal dimensions must expand from their initial size close tothe fundamental Planck length M −1

∗to the stabilization value b0 10−17+30 /n cm leads

to natural inationary scenarios involving the dynamics of the internal space. Theresolution of cosmological conundrums such as the horizon, atness and age problems,and the production of the spectrum of nearly-scale invariant Harrison-Zeldovich densityperturbations with the avoidance of drastic ne tuning of the inaton mass comeas a consequence of the evolution of the internal space. Moreover it is remarkablethat the era of post-ination brane-contraction that follows this period of inationis harmless, and automatically ends via a “Big Bounce”. During the phase of b(t)

evolution to the stabilization point, the production of bulk gravitons by the time-varying metric remains completely suppressed, ensuring that the bulk is very coldat, and after, the stabilization of the internal dimensions. The primary remainingissue is the radion moduli problem, which is no more severe than in gauge-mediatedsupersymmetry breaking models. Overall, then, early universe cosmology in thesemodels is quite interesting!

Acknowledgments

It is a pleasure to thank Gia Dvali for valuable discussions of related ideas [ 25] and toAndrei Linde for useful conversations. SD thanks the CERN theory group, and JMRthanks the Stanford University theory group, for their respective hospitality duringportions of this work. The work of SD and NK is supported in part by NSF grant

34



PHY-9870115. The work of JMR is supported in part by an A.P. Sloan FoundationFellowship.

Appendix A: Kinematics of the radion eld

With the metric of the form ( 9) the Ricci scalar is

− R= 6aa

+ 6aa

2

+ 2 nbb

+ n(n −1)bb

2

+ 6 nabab

+κn (n −1)

b2 , (104)

where the internal curvature term is present for n-spheres (κ = 1), but vanishes fortori ( κ = 0), and we have ignored a similar curvature term for the large dimensions.After integrating over all spatial coordinates we obtain,

S = dt(LKE (a, b) −a3V (b)), (105)

and further integrating the ¨ a and b terms by parts, the kinetic part of the action forthe radii, a and b, becomes

S = −M 2+ n∗ dt a 3bn 6

aa

2

+ n(n −1)bb

2

+ 6 nabab

. (106)

Note the overall negative sign of these kinetic terms, but also the mixing between ˙ aand b.

There is clearly an extremum of the action with ˙a = b = 0, when the condition∂ a (a3V tot (b))|a= a 0 ,b= b0 = 0, and similar with ∂ a →∂ b are met. These imply (for a0 = 0)

V tot (b0) = 0 , andV tot (b0) = 0 . (107)

This is as one would have naively expected. However, because of the negative signfor the kinetic term for the radial degrees of freedom, the stability analysis for suchstatic solutions has to be treated with care. The analysis starts by expanding theaction, Eq. ( 106), in small uctuations around the extremum: a(t) = a0 + δa(t), andb(t) = b0 + δb(t). Then to quadratic order, and dening ∆ ≡δa/a 0 and δ ≡δb/b0, theexpansion gives the coupled equations of motion

6 3n3n n(n −1)

∆δ

= 0 00 ω2

∆δ , (108)

35



whereω2 =

12

(b0)2V tot (b0)M 2+ n∗

(b0)n =12

(b0)2V tot (b0)M 2pl

. (109)

is the radion mass around the stabilization point. Searching for oscillating solutions,(∆ , δ) = exp( iΩt)(∆ 0, δ0) of the stability equations leads to an eigenvalue problem forthe frequency Ω. Specically, Ω2 has the eigenvalues Ω2 = 0, and

Ω2 =2

n(2 + n)ω2. (110)

The zero eigenvalue just corresponds to the fact that a0 is a at direction since, byassumption, there is no potential for a. The crucial expression is Eq. ( 110), which givesthe condition for stability of the static solution. In the end, stability just requires thatthe radion (mass) 2 be positive as one would expect, and that we can think in terms of a total potential V (b) that one can minimize to nd the stable static solutions for thesize of the internal dimensions.

The equations of motion for a(t) and b(t) derived from the action Eqs. ( 105) and (106)are, after some algebra, those given in the text, namely ( 10), (11) and the constraint(12) which comes about from the well-known property that the total energy in GR iszero. (This constraint can be derived by carefully working in terms of the lapse andshift functions of the canonical formalism.) If matter on our wall is also included thenthey become those given in ( 57):

6H 2a + n(n

−1)H 2b + 6 nH a H b =

V + ρ

M n +2

∗ bn

bb

+ ( n −1)H 2b + 3 H bH a =1

M n +2∗

bn2V

n + 2 −b

n(n + 2)∂V ∂b

+ρ −3 p

2(n + 2)aa

+ 2 H 2a + nH bH a =1

M n +2∗

bnb

2(n + 2)∂V ∂b −

n −22(n + 2)

V +ρ + ( n −1) p

2(n + 2)ρ + 3 H a ( p + ρ) = 0 . (111)

Note that in these equations the effect of wall-localized matter is just some extracontribution to the a(t) and b(t) scale-factor evolution. Of course the energy densityon the brane in general distorts the geometry of the internal space (as does that presenton other branes that may exist in the bulk), but as far as the overall properties andevolution of the zero mode size modulus b(t) of the internal space is concerned, it iscorrect to treat the energy density on the wall as just averaged over the whole space,as done on the RHS of these equations.

36



It is useful to summarize some basic properties of the evolution equations. Since thetotal potential energy U (a,b,ψ) is given in terms of the effective 4-dimensional potentialenergy density V (b, ψ) by U = a3V (b, ψ), a uniform bulk cosmological constant isrepresented by V = bn Λ. Substituting this form into the RHS of these equations we

see that a positive bulk cosmological constant term has the effect of wanting to increaseboth a and b if the evolution of a, b are studied close to zero. This is not inconsistentwith the stability criteria derived above since this was explicitly the stability analysisaround a stationary point of the equations with non-zero values of both a and b.Indeed, the stability analysis can be derived directly from the equations of motion, asit must. Concretely, if we expand around a point ( a

∗, b∗) with V (b

∗) = V (b

∗) = 0,

then linearizing the equations of motion gives

δaδb =

0 a∗b1− n∗

2(n +2) M n +2∗

V

0

−b2− n∗

n (n +2) M n +2

∗

V δaδb , (112)

which exactly reproduces the previous stability analysis, in particular the requirementV (b

∗) > 0 for a

∗, b∗

> 0.Another basic property that is useful to keep in mind is the effect of some small

amount of wall-localized matter on the position and mass of the radion. Linearizingthe equations ( 57) with such matter in the shift δb = δb/b0 around the stabilizationpoint we nd

δb + 3 δbH a = −1

(n + 2) bn0 M n +2

∗

b20V n

+n(ρ −3 p)

2δb +

(ρ −3 p)2(n + 2) bn

0 M n +2

∗

. (113)

This shows that wall-localized matter has two effects o the radion. First, the δb-independent term on the RHS shifts the radion from the stabilization point by a smallamount, and second the ( ρ, p)-dependent in the parentheses on the RHS shifts theeigenfrequency of oscillations of the radion (or equivalently the radion mass). Boththese effects are to be expected, and are harmless for wall-localized matter densitiesand pressures ρ, p M 4

∗.

Appendix B: Kasner-like solutions with potential

The solutions of (68) take different form, controlled by whether C 2 vanishes or notand the sign of ∆. Let us rst consider the case ∆ > 0. Then, when C 2 = 0, theequation ( 68) simplies to X = ± 2 ∆

n (n +2) ω exp(X/ 2), which can be integrated togive exp(X ) = 2n (n +2)

∆ ω1

τ 2 . Here we have removed an additional integration constant

37



by a time translation. It is straightforward now to use this and ( 63) to determine thesolutions for α and β . Since α = (2n + p)( n− p)C 1

4∆ −n (n + p−2)2∆ X and β = 2n + p

∆ X −3(2 n + p)C 12∆ ,

we nd

eα

= exp(2n + p)(n

− p)C 1

4∆∆ ω

2n(n + 2)

n ( n + p − 2)2∆

τ n ( n + p − 2)

∆

eβ = exp −3(2n + p)C 1

2∆2n(n + 2)

∆ ω

2n + p∆ 1

τ 4n +2 p

∆. (114)

We can now use (61) to transform ( 114) back to using the comoving time on the brane.Integrating ( 61), we nd

τ ∼t−∆

( n + p )(2 n + p ) . (115)

Note the important feature of this map, that it maps the comoving future t → ∞to theorigin of “time” τ and vice-versa. Thus to discover the correct long-time asymptoticsof the solutions we must extract the small- τ behavior. This is true for the general caseC 2 = 0 as well.

In any case, for C 2 = 0, and after appropriate rescalings, we nd the behavior of thescale factors a(t) and b(t) to be

a = a itt i

−n ( n + p − 2)

( n + p )(2 n + p )

b = bitt i

2n + p

. (116)

These solutions describe the cases where the brane length scales shrink while the radioncontinues to grow. This means that even if the brane scale factor a(t) continues toexpand by inertia after the end of ination, soon after a(t) starts to contract. Moreover,the behavior ( 116) is an asymptotic attractor for the generic cases with C 2 < 0 and∆ > 0, as we will now discuss.

Let us now consider the case when C 2 < 0. Again, note from (63) that this corre-sponds to the situation where initially both H a and H b are positive, when ination has just ended. The equation ( 68) can be solved in this case by the substitution

e−X = 8∆ ω3(2n + p)2C 22

sinh2(ϑ) (117)

which reduces to the differential equation ϑ = 3(2 n + p)2

16n (n +2)

1/ 2C 2, whose solution is ϑ =

3(2 n + p)2

16n (n +2)

1/ 2C 2τ , after the appropriate choice of the coordinate origin of τ . Therefore,

38



the solution of ( 68) is

e−X =8∆ ω

3(2n + p)2C 22sinh2( 3(2n + p)2

16n(n + 2)C 2τ ) (118)

Note that the RHS is a positive semidenite function, guaranteeing the reality of themetric, as required. Also note that the solution ( 118) is dened in open intervalτ ∈(0+ , ∞) and (−∞, 0−), by time-reversal. Using this equation and ( 63), we canextract the solutions for α and β :

eα =8∆ ω

3(2n + p)2C 22

n ( n + p − 2)∆ e

(2 n + p )( n − p )4∆ (C 1 + C 2 τ ) sinh

n ( n + p − 2)∆ ( 3(2n + p)2

16n(n + 2) |C 2τ |)

eβ =3(2n + p)2C 22

8∆ ω

2n + p∆

exp −3(2 n + p)

2∆ (C 1 + C 2τ )

sinh2 2n + p∆ ( 3(2 n + p)2

16n (n +2) |C 2τ |)(119)

where the argument of the hyperbolic sine is taken to be positive, to make sure that thesolutions remain real. In general, it is not possible to explicitly determine the integratedform of the gauge transformation ( 61). However, the asymptotic limits τ →0, ∞areeasy to deduce, and are sufficient for our purpose here.

Taking the τ →0 limits of (119) we nd that ( 119) reduce precisely to ( 114),implying that the integrated form of ( 61) approaches ( 115). Hence in the case C 2 < 0the scale factors again approach ( 69) as t → ∞, which are therefore the appropriatefuture attractors in all cases ∆ > 0, C 2 ≤0.

It is also amusing to consider the short t-time behavior of the exact solutions in thiscase. We nd that as τ → ∞,eα∼ exp

(2n + p)C 2τ 4∆

(n − p−n(n + p −2) 3n(n + 2)

)

eβ ∼ exp −

(2n + p)C 2τ 2∆

(3 −(2n + p) 3n(n + 2)

) (120)

Using this, we can integrate ( 61) and nd

t∼exp(2n + p)C 2τ

4∆(n(n − p + 6) 3

n(n + 2) −3(n + p)) (121)

and upon substituting this back into ( 120), we can show that the solutions approachfrom below the asymptotic expressions

eα →tt i

3+ √3n ( n +23( n +3)

39



eβ →tt i

n − √3n ( n +2)n ( n +3) (122)

The dependence on the parameter p apparent in ( 120) has completely disappeared.This can be most easily seen by taking the derivatives with respect to p of the powers

which are obtained by substituting ( 121) into (120), noting that they are identicallyzero, and then setting p = 0 to obtain ( 122). Also note that the resulting powers areidentical to those found in the simple Kasner case ( 54), with lower sign taken. Thusin the very early time limit, this suggests that the universe was expanding while theradion was decreasing. However, this phase is cut out by a stage of ination, and infact only a very short portion is retained where both a and b are growing for shorttime, to match onto the post-inationary era.

Now we consider the last case, ∆ < 0, which we argued in the text should bedescribed by the traditional potential-free Kasner solutions in the long-time limit. We

can see this explicitly by examining the exact solutions once again. First note that,in this case, C 2 must be nonzero, as can be immediately seen from ( 68). The exactsolutions can be found by the substitution

e−X =8|∆ |ω

3(2n + p)2C 22cosh2(ϑ) (123)

which is analogous to (117). The equation for ϑ is the same as before, and hence thesolution is

e−X =8|∆ |ω

3(2n + p)2

C 2

2

cosh2 3(2n + p)2

16n(n + 2)

C 2τ (124)

The similarity of this solution to ( 118) allows us to extract the expressions for α andβ quite easily. We nd

eα =8|∆ |ω

3(2n + p)2C 22

n ( n + p − 2)| ∆ | e

(2 n + p )( n − p )4 | ∆ | (C 1 + C 2 τ ) cosh

n ( n + p − 2)| ∆ | 3(2n + p)2

16n(n + 2) |C 2τ |

eβ =3(2n + p)2C 22

8|∆ |ω2n + p

| ∆ |exp −

3(2 n + p)2|∆ | (C 1 + C 2τ )

cosh2 2n + p| ∆ | 3(2 n + p)2

16n (n +2) |C 2τ |(125)

Note however the important difference between these solutions and ( 119). In this case,the solutions are dened on the whole interval ( −∞, ∞). By considering the limitsτ → ±∞, we can see that the solutions ( 125) in fact interpolate between the simpleKasner solutions ( 52), where for C 2 < 0 it starts out with powers given by ( 54) withlower sign and transmutes due to the intermediate potential-dominated region into ( 52)

40



with powers (54) with upper sign taken. In reality, the far past of these solutions iscut out by the inationary era, and again only a very short portion where both a andb are growing is retained, which is matched onto the post-inationary era. All of thesesolutions therefore ow towards simple Kasner solutions with the powers with upper

sign in (54), which are the appropriate future attractors.

Appendix C: Exact Solutions for the Big Bounce

To see that bounce behavior does indeed occur in the presence of radiation on the wallwe go back to our equations of motion but now set the potential to zero and just keepthe wall-localized radiation terms on the RHS:

6H 2a + n(n −1)H 2b + 6 nH a H b =ρ

M n +2∗

bn

bb

+ ( n −1)H 2b + 3 H bH a = ρ −3 p2(n + 2) M n +2

∗bn

aa

+ 2 H 2a + nH bH a =1

M n +2∗

bnρ + ( n −1) p

2(n + 2)ρ + 3 H a ( p + ρ) = 0 . (126)

These can again be solved exactly, and provide a good approximation for the exit fromthe Kasner-like phase of a(t)-contraction.

Set p = ρ/ 3 for radiation. The solution for the radiation energy density is as usual

ρ = BM n +2∗

a4 (127)

(with some normalization B ) and dene variables α and β by

a = a0eα

b = b0eβ (128)

where a0 and b0 are constants, determined by the values of a and b at the end epochwhen the radion potential becomes small compared to the wall-localized radiation.

Using this, the equations of motion can be rewritten as6 α2 + n(n −1)β 2 + 6 nα β = Ae−nβ −4α

β + β (3α + nβ ) = 0

α + α (3α + nβ ) =A6

e−4α−nβ (129)

41



where A = B/ (a40bn

0 ). If we change the time variable, dening the new time τ as in eq.(61):

dτ = −e−3α −nβ dt (130)

where again the specic form of τ = τ (t) can be found later when solutions are deter-

mined, we can rewrite the equations of motion as

6α 2 + n(n −1)β 2 + 6 nα β = Ae2α + nβ

β = 0

α =A6

e2α + nβ (131)

where the primes denote derivatives with respect to τ . The β equation immediatelygives

β = C 1 + C 2τ (132)

Hence note that if H b > 0 initially then the constant C 2 < 0. Also note that given ourconventions we again have future t innity map to τ = 0.

For the purpose of examining the solutions to these equations we distinguish twocases:

(i) C 2 = 0; like before, this solution turns out to be a future attractor of all solutions !

This case has a very simple analysis: C 2 = 0 implies β = 0 but also β = 0. Hencewe can forget about the τ coordinates and immediately work in our original brane-timet. So b = 0 means b = b0 = const and so the constraint equation gives

6H 2a =A

bn0 a4 (133)

which is immediately solved to give us the radiation dominated universe, a∼√t. This

is precisely as we would expect from an analysis in the Einstein frame.

(ii) C 2 = 0; this case corresponds to more generic solutions. In particular we willsee that we can have initial conditions H b > 0, H a < 0, which evolve to H b = 0,

H a > 0, in other words a “bounce” solution.

To solve the equations in this case, rst dene

X = 2α + nβ, (134)

42



in terms of which, the second order differential equations produce

X =A3

eX (135)

This is again the Liouville equation, with the rst integral (conservation of energy)

X 2 =2A3

eX + Z 0 (136)

where Z 0 is another integration constant. If we eliminate eX from the constraintequation by using the last equation, and then simplify the result by using the secondequation, we nd that the constraint reduces to

Z 0 =n(n + 2)

3C 22 ≥0 (137)

Thus the constant C 2 controls the dynamics, and the whole system has collapsed down

to two simple equations, one of which is already solved:β = C 1 + C 2τ

X 2 =2A3

eX +n(n + 2)

3C 22 . (138)

Note that the solution to the X equation is,

eX =n(n + 2) C 22

2A1

sinh2( n (n +2)12 C 2τ )

(139)

as may be checked by substitution. Using the formula for X = 2α + nβ and β =C 1 + C 2τ , we nd that

eα = +n(n + 2) C 22

2A

1/ 2 e−nC 1 / 2+ n |C 2 |τ / 2

sinh( n (n +2)12 |C 2|τ )

, (140)

where since C 2 < 0 in the cases of interest we have taken the appropriate branch of the square-root such that a = a0eα is positive as it must be.

This expression is already sufficient to show that we get a bounce behavior for a(t).Recognizing that t = 0 corresponds to τ → ∞while t → ∞corresponds to τ →0,simply plotting eα as given by (140) shows that a(t) goes through a bounce.

It is instructive to see this in detail. First, consider the limit τ →0. Clearly,β = C 1 + C 2τ →C 1 = const . Also,

eα →+6e−nC 1

A

1/ 2 1τ

(141)

43



Thus exp(3 α + nβ ) →τ −3, and so we have

dt∼−dτ τ 3

(142)

or, t

∼

τ −2 which gives us τ

∼

1/ √t, and t

→ ∞maps to τ

→0+ as claimed. Back in

the formula for a, this gives

a∼exp(α)∼1τ ∼

√t (143)

and this is precisely a radiation-dominated universe at late times! So indeed, thesolutions with C 2 = 0 are late time attractors. Some further analysis shows that inthis limit, (and in the approximation of ignoring the stabilizing potential) b will tendto a constant logarithmically. The end result of the analysis is that for large t thesesolutions show that a is expanding asymptotically as √t .

Let’s now look at the other limit, τ → ∞. Also, recall that C 2 must be less thanzero: C 2 < 0. We have

b∼exp(β )∼exp(−|C 2|τ ) (144)

anda∼exp(α)∼exp |C 2|τ

2n − n(n + 2) / 3 . (145)

These give the relationship between τ and t

dt

∼−exp |C 2|τ

2n

− 3n(n + 2) dτ (146)

Now note that ( n − 3n(n + 2)) < 0 for all 2≤n ≤6, and therefore as τ →+ ∞wehave t →0+ as

t∼exp −|C 2|τ 2

( 3n(n + 2) −n) (147)

as claimed. Finally, in the expression ( 145) for a(t), note that ( n − n(n + 2) / 3) > 0,so as τ → ∞, the scale factor a(t) is again going to innity. Re-expressed in terms of t we see that as t initially increases a(t) is initially decreasing .

If one carefully considers this limit the power law behavior of a and b in terms of thetime t precisely corresponds to the late-time Kasner solutions we found in Appendix B.Most importantly, we see that as τ decreases from ∞ towards 0 + , the radion b isinitially increasing and a is initially decreasing, the initial conditions that we require.But as we have seen from the analysis in the τ →0 limit, this goes over to b growing

44



logarithmically and a increasing as 1/τ . This implies there must indeed have been aBig Bounce in between!

Finally, the bounce occurs when α = 0 or equivalently H a = 0. This happens when

n(n

−1)H 2

b=

ρwall

M n +2∗ bn

. (148)

Hence, when the bounce occurs the radion kinetic energy is comparable to the wallenergy density. By continuity, it should be clear that the generic qualitative featuresof these properties would remain true even in the presence of stabilizing potentials.The main conclusion of this analysis is that the Big Bounce is the future asymptoticattractor of all postinationary solutions with wall radiation, and hence the exit fromthe contraction on the wall will occur naturally.

References[1] N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. B429 , 263 (1998).

[2] I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. B436 ,257 (1998).

[3] N. Arkani-Hamed, S. Dimopoulos and G. Dvali, hep-ph/9807344 .

[4] N. Arkani-Hamed, S. Dimopoulos and J. March-Russell, hep-th/9809124 .

[5] J. Polchinski, Phys. Rev. Lett. 75 (1995) 4724;Also see for example: J. Polchinski, TASI lectures on D-branes , hep-th/9611050 ;C. Bachas, Lectures on D-branes , hep-th/9806199 .

[6] K. Dienes, E. Dudas and T. Gherghetta, Phys. Lett. B436 , 55 (1998); Nucl. Phys.B537 (1999) 47.

[7] I. Antoniadis, Phys. Lett. B246 (1990) 377;I. Antoniadis and K. Benakli, Phys. Lett. B326 (1994) 69;I. Antoniadis, K. Benakli and M. Quiros, Phys. Lett. B331 (1994) 313.

[8] P. Horava and E. Witten, Nucl. Phys. B460 (1996) 506;E. Witten Nucl. Phys. B471 (1996) 135;J.D. Lykken, Phys. Rev. D54 (1996) 3693;E. Caceres, V.S. Kaplunovsky and I.M. Mandelberg, Nucl. Phys. B493 (1997) 73.

45

http://xxx.lanl.gov/abs/hep-ph/9807344

http://xxx.lanl.gov/abs/hep-th/9809124









[9] G. Shiu and S.H. Tye, Phys. Rev. D58 , 106007 (1998).

[10] C. Bachas, hep-ph/9807415 ;I. Antoniadis and C. Bachas, hep-th/9812093 .

[11] R. Sundrum, hep-ph/9805471 .

[12] R. Sundrum, hep-ph/9807348 .

[13] P. Argyres, S. Dimopoulos and J. March-Russell, hep-th/9808138 ;Z. Kakushadze and S.-H. Tye, hep-th/9809147 ;K. Dienes, et al. , hep-ph/9809406 ;K. Benakli, hep-ph/9809582 ;L. Randall and R. Sundrum, hep-th/9810155 ;G.F. Giudice, R. Rattazzi and J.D. Wells, hep-ph/9811291 ;

S. Nussinov and R. Shrock, hep-ph/9811323 ;E.A. Mirabelli, M. Perelstein and M.E. Peskin, hep-ph/9811337 ;T. Han, J.D. Lykken and R. Zhang, hep-ph/9811350 ;J.L. Hewett, hep-ph/9811356 ;Z. Berezhiani and G. Dvali, hep-ph/9811378 ;K.R. Dienes, E. Dudas and T. Gherghetta, hep-ph/9811428 ;N. Arkani-Hamed, et al. , hep-ph/9811448 ;Z. Kakushadze, hep-th/9812163 .

[14] K. Benakli and S. Davidson, hep-ph/9810280 .[15] D. Lyth, hep-ph/9810320 .

[16] N. Kaloper and A. Linde, hep-th/9811141 .

[17] G. Dvali and S.H. Tye, hep-ph/9812483 .

[18] A. Mazumdar, hep-ph/9902381 .

[19] A. Linde, Phys. Lett. B129 , 177 (1983).

[20] A. D. Linde, Particle Physics and Inationary Cosmology (Harwood, Chur,Switzerland, 1990).

[21] See for example:C. Mathiazhagan and V.B. Johri, Class. Quant. Grav. 1 , L29 (1984);

46

















































F. Lucchin and S. Matarrese, Phys. Rev. D32 (1985) 1316;A.B. Burd and J.D. Barrow, Nucl. Phys. B308 (1988) 929.

[22] S. Kalara, N. Kaloper and K.A. Olive, Nucl. Phys. B341 (1990) 252.

[23] L. Ford, Phys. Rev. D35 (1987) 2955.

[24] See: L. Parker, in Asymptotic Structure of Space-Time , Cincinnati 1976, eds.F. Esposito and L. Witten, (Plenum, New York);Y.B. Zeldovich, JETP Lett. 12 (1970) 307.

[25] G. Dvali, in preparation.

Nima Arkani-Hamed et al- Rapid asymmetric inflation and early cosmology in theories with...

Documents

Transcript of Nima Arkani-Hamed et al- Rapid asymmetric inflation and early cosmology in theories with...